Finite Group Databasebeta

← #27181074

Subgroups — #27181074

Order 12928.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692230C211yesmaximal, characteristic
2#117×D101C211yesmaximal, characteristic, complemented
3#20265504C212yesmaximal, complemented
4#20265503C212yesmaximal, complemented
5#20265497C212yesmaximal, complemented
6#20265492C212yesmaximal, complemented
7#20265217C211yesmaximal, characteristic, complemented
8#20265183C211yesmaximal, characteristic, complemented
9#20265181C211yesmaximal, characteristic, complemented
10#20265140C211yesmaximal, characteristic, complemented
11#428×C101C211yesmaximal, characteristic, complemented
12C22×D404—21no—
13C22×D404C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15C2×#13108058C2211yescharacteristic
16#15692230—21no—
17#15692230—21no—
18#15692230C2212yes—
19#15692230C2212yes—
20#15692214C2211yescharacteristic
21#15692213C2211yescharacteristic, complemented
22#15692213C2211yescharacteristic, complemented
23#15692212—21no—
24#15692211—21no—
25C4⋊C4×D101C2211yescharacteristic, complemented
26C4⋊C4×D101C2211yescharacteristic, complemented
27C2×#13108051—21no—
28C22⋊C4×D101—21no—
29C42×D101—21no—
30#15692188C2212yescomplemented
31#15692186C2212yescomplemented
32#15692177C2212yescomplemented
33#15692177C2212yescomplemented
34#15692175C2212yescomplemented
35#15692175C2212yescomplemented
36#15692167C2212yescomplemented
37#15692167C2212yescomplemented
38C2×M16×C101C2211yescharacteristic
39#117×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#107×C101C2212yescomplemented
42#106×C101C2212yescomplemented
43C23×D101—21no—
44C23×D101D411yescharacteristic
45C2×D404—22no—
46C2×D404—22no—
47C2×D404—22no—
48C2×D404—22no—
49C2×D404—21no—
50C2×D404—44no—
51C2×D404C2312yes—
52C22×Dic101D411yescharacteristic
53C2×C4×D101—22no—
54C2×C4×D101—22no—
55C2×C4×D101—21no—
56C2×C4×D101—21no—
57C2×C4×D101D411yescharacteristic
58C2×C4×D101D411yescharacteristic
59C2×C4×D101C2311yescharacteristic
60C2×C4×D101C2311yescharacteristic
61#13108058—22no—
62C2×#11717579C2312yes—
63C4⋊Dic101C2311yescharacteristic
64C4⋊Dic101C2311yescharacteristic
65#13108053—21no—
66#13108053—21no—
67C22⋊Dic101—21no—
68#13108051—21no—
69#13108051—21no—
70#13108051—22no—
71#13108051—22no—
72C4×Dic101—21no—
73C4×Dic101—21no—
74C4×Dic101—21no—
75C22×C4×C101—21no—
76C22×C4×C101C2311yescharacteristic
77M16×C101—22no—
78C2×C8×C101C2312yes—
79C4⋊C4×C101—21no—
80C4⋊C4×C101C2311yescharacteristic
81C4⋊C4×C101C2312yes—
82C4⋊C4×C101C2311yescharacteristic
83C22⋊C4×C101—21no—
84C42×C101—21no—
85C23×C101C2×D411yescharacteristic
86C22×D101—22no—
87C22×D101—22no—
88C22×D101—22no—
89C22×D101—41no—
90C22×D101—42no—
91C22×D101—42no—
92C22×D101—42no—
93C22×D101C2×D411yescharacteristic
94C22×D101C2×D411yescharacteristic
95C2×C4×C101—21no—
96C2×C4×C101—21no—
97C2×C4×C101—22no—
98C2×C4×C101—42no—
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101#4112yes—
101C2×C4×C101#4112yes—
102C2×C4×C101C2411yescharacteristic
103C2×Dic101—22no—
104C2×Dic101—21no—
105C2×Dic101—21no—
106C2×Dic101C2×D411yescharacteristic
107C2×Dic101C2×D411yescharacteristic
108D404—22no—
109D404—22no—
110D404—44no—
111D404—44no—
112D404—42no—
113C4×D101—22no—
114C4×D101—22no—
115C4×D101—21no—
116C4×D101—21no—
117C4×D101—21no—
118C4×D101—21no—
119C4×D101—42no—
120C4×D101—42no—
121C8×C101—22no—
122#11717579—22no—
123C22×C101—21no—
124C22×C101—21no—
125C22×C101#12312yes—
126C22×C101C22×D411yescharacteristic
127C2×D101—21no—
128C2×D101—21no—
129C2×D101—21no—
130C2×D101—21no—
131C2×D101—21no—
132C2×D101—44no—
133C2×D101—42no—
134C2×D101—42no—
135C2×D101—42no—
136C2×D101—84no—
137C2×D101AGL(1,ℤ/8)12yes—
138C4×C101—41no—
139C4×C101—42no—
140C4×C101—41no—
141C4×C101#12312yes—
142C4×C101C2×#4111yescharacteristic
143C4×C101C2×#4111yescharacteristic
144Dic101—21no—
145Dic101—21no—
146Dic101—41no—
147Dic101—41no—
148C2×C101—22no—
149C2×C101C2×#12311yescharacteristic
150C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C101#57511yescharacteristic
152D101—22no—
153D101—41no—
154D101—82no—
155#3094—1011nomaximal, Sylow
156C101#309411yescharacteristic, complemented, Sylow
157C2×#121—1011no—
158C2×#117—1011no—
159#482—1012no—
160#482—1012no—
161#481—1012no—
162#480—1012no—
163#428—1011no—
164#428D10111yescharacteristic, complemented
165#418—1011no—
166#418—1011no—
167#407—1011no—
168C22×D4—1011no—
169C22×D4—2021no—
170C23×C4—1011no—
171C2×M16—1011no—
172C2×M16C2×D10111yescharacteristic
173#121—1012no—
174#121—1012no—
175#121—2021no—
176#121—2021no—
177#117—1011no—
178#117—1011no—
179#117—1011no—
180#117—2021no—
181#117—2021no—
182#117C2×D10111yescharacteristic
183C2×C4⋊C4—1011no—
184C2×C4⋊C4—1011no—
185C2×C4⋊C4C2×D10111yescharacteristic
186C2×C22⋊C4—2021no—
187C2×C22⋊C4—2021no—
188C2×C42—2021no—
189#107—1012no—
190#107C2×D10112yescomplemented
191#106—1012no—
192#106C2×D10112yescomplemented
193#102—1012no—
194#102—1012no—
195#102—1012no—
196#102—1012no—
197C22⋊C8—1012no—
198C22⋊C8—1012no—
199C24—1011no—
200C24—2021no—
201C2×D4—1012no—
202C2×D4—4044no—
203C2×D4—2022no—
204C2×D4—2022no—
205C2×D4—2021no—
206C2×D4—2022no—
207C2×D4—2022no—
208C22×C4—21no—
209C22×C4—1011no—
210C22×C4—1011no—
211C22×C4—1011no—
212C22×C4—1011no—
213C22×C4—1011no—
214C22×C4—2021no—
215C22×C4—2021no—
216C22×C4—2022no—
217C22×C4—2022no—
218C22×C4C22×D10111yescharacteristic
219M16—22no—
220M16—2022no—
221C2×C8—1012no—
222C2×C8C22×D10112yes—
223C4⋊C4—21no—
224C4⋊C4—1011no—
225C4⋊C4—1011no—
226C4⋊C4—2021no—
227C4⋊C4—2021no—
228C4⋊C4C22×D10111yescharacteristic
229C4⋊C4C22×D10111yescharacteristic
230C4⋊C4C22×D10112yes—
231C22⋊C4—21no—
232C22⋊C4—2022no—
233C22⋊C4—2022no—
234C22⋊C4—2021no—
235C22⋊C4—2021no—
236C22⋊C4—2021no—
237C42—21no—
238C42—2021no—
239C42—2021no—
240C42—2021no—
241C23—1011no—
242C23—1011no—
243C23—4042no—
244C23—4042no—
245C23—4041no—
246C23—4042no—
247C23—2022no—
248C23—2022no—
249C23—2022no—
250C23D4×D10111yescharacteristic
251D4—4044no—
252D4—4044no—
253D4—4042no—
254D4—2022no—
255D4—2022no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—1011no—
261C2×C4—1011no—
262C2×C4—4042no—
263C2×C4—4042no—
264C2×C4—2021no—
265C2×C4—2021no—
266C2×C4—2022no—
267C2×C4—2021no—
268C2×C4—2021no—
269C2×C4—2021no—
270C2×C4—2021no—
271C2×C4—2022no—
272C2×C4—2022no—
273C2×C4D4×D10111yescharacteristic
274C2×C4#1310807412yes—
275C2×C4#1310807512yes—
276C2×C4C23×D10111yescharacteristic
277C8—22no—
278C8—2022no—
279C22—21no—
280C22—21no—
281C22—8084no—
282C22—1012no—
283C22—4044no—
284C22—4042no—
285C22—4042no—
286C22—4042no—
287C22—2021no—
288C22—2021no—
289C22—2021no—
290C22—2021no—
291C22—2021no—
292C22#1569224012yes—
293C22C2×D4×D10111yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—4041no—
298C4—4041no—
299C4—2021no—
300C4—2021no—
301C4#1569224012yes—
302C4C2×#1310807411yescharacteristic
303C4C2×#1310807511yescharacteristic
304C2—22no—
305C2—8082no—
306C2—4041no—
307C2—2022no—
308C2C2×#1569224011yescharacteristic
309C2AGL(1,ℤ/8)×D10111yescharacteristic
310C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692230C2
hypercenter#428D101
hypocenterC101#3094
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #117×D101C2semidirect
3. #20265504C2semidirect
4. #20265503C2semidirect
5. #20265497C2semidirect
6. #20265492C2semidirect
7. #20265217C2semidirect
8. #20265183C2semidirect
9. #20265181C2semidirect
10. #20265140C2semidirect
11. #428×C101C2semidirect
21. #15692213C22semidirect
22. #15692213C22semidirect
25. C4⋊C4×D101C22semidirect
26. C4⋊C4×D101C22semidirect
30. #15692188C22semidirect
31. #15692186C22semidirect
32. #15692177C22semidirect
33. #15692177C22semidirect
34. #15692175C22semidirect
35. #15692175C22semidirect
36. #15692167C22semidirect
37. #15692167C22semidirect
41. #107×C101C22semidirect
42. #106×C101C22semidirect
156. C101#3094semidirect
164. #428D101semidirect
190. #107C2×D101semidirect
192. #106C2×D101semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156922302
#117×D1012
#202655042
#202655032
#202654972
#202654922
#202652172
#202651832
#202651812
#202651402
#428×C1012
#3094101